Transreal arithmetic as a consistent basis for paraconsistent logics Conference or Workshop Item 

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Anderson, J. A.D.W. and Gomide, W. (2014) Transreal arithmetic as a consistent basis for paraconsistent logics. In: International Conference on Computer Science and Applications (WCECS 2014), 22­24 October 2014, San Francisco, USA, , pp. 103­108. Available at http://centaur.reading.ac.uk/37404/ 

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Transreal Arithmetic as a Consistent Basis

For Paraconsistent Logics

James A.D.W. Anderson and Walter Gomide

Author Final Version: July 25, 2014

J.A.D.W. Anderson is with the School of Systems Engineering, ReadingUniversity, England, RG6 6AY e-mail: j.anderson@reading.ac.uk

Walter Gomide is with the Federal University of Mato Grosso (UFMT),Brazil e-mail: waltergomide@yahoo.com

Abstract

Paraconsistent logics are non-classical logics which allow non-trivialand consistent reasoning about inconsistent axioms. They have been pro-posed as a formal basis for handling inconsistent data, as commonly arisein human enterprises, and as methods for fuzzy reasoning, with applica-tions in Artificial Intelligence and the control of complex systems.

Formalisations of paraconsistent logics usually require heroic mathe-matical efforts to provide a consistent axiomatisation of an inconsistentsystem. Here we use transreal arithmetic, which is known to be consis-tent, to arithmetise a paraconsistent logic. This is theoretically simpleand should lead to efficient computer implementations.

We introduce the metalogical principle of monotonicity which is a verysimple way of making logics paraconsistent.

Our logic has dialetheaic truth values which are both False and True.It allows contradictory propositions, allows variable contradictions, butblocks literal contradictions. Thus literal reasoning, in this logic, forms anon-the-fly, syntactic partition of the propositions into internally consistentsets. We show how the set of all paraconsistent, possible worlds can berepresented in a transreal space. During the development of our logic wediscuss how other paraconsistent logics could be arithmetised in transrealarithmetic.

Keywords: transreal arithmetic, paraconsistent logic.

1 Introduction

Paraconsistent logics were explicitly introduced in the second half of the twen-tieth century as non-classical logics that can reason about inconsistent axioms[26][13]. In a classical logic, inconsistent axioms explode, allowing any theoremto be proved in a trivial way [11]. By contrast paraconsistent logics do not

explode, they allow only limited conclusions to be drawn from inconsistent ax-ioms. Some paraconsistent logics admit dialetheias, that is propositions whichare both False and True [24], and some admit Gap values with no component offalsity or truthfulness [29]. Gap values are usually treated absorptively so thatany logical combination with a Gap produces a Gap as result. This behaviouris consistent with one reading [20] of Frege’s principle of compositionallity sothat a compound proposition lacks reference if any component of it lacks refer-ence. It should be added that paraconsistent logics are also capable of classicalreasoning so they provide a robust generalisation of classical logic. This makesthem interesting both from a theoretical and a practical perspective.

It has been proposed that paraconsistent logics are suitable for implementingfuzzy logical systems, including the control of complex systems [34][10], for prac-tical reasoning about inconsistent data, such as the data typically provided byhumans, for use in Artificial Intelligence programs, for human use in developingscientific theories that contain contradictory elements, and more [26].

Paraconsistent logics are usually formalised in terms of advanced mathe-matics. Shramko [29] reviews some logics using small, discrete, ordered sets oftruth values; these are mathematically simple, though one of the logics uses twoorderings. Logics with more than two orderings are discussed in [30]. Someparaconsistent logics partition propositions into a hierarchy of internally consis-tent partitions.

Here we take the simpler approach of expressing a paraconsistent logic arith-metically. We use transreal arithmetic, which is a generalisation of real arith-metic. Transreal arithmetic was originally developed [2][3] from a subset of thealgorithms used in the arithmetic of fractions. It has been axiomatised and amachine proof of consistency has been given [8]. Two human proofs of consis-tency are known but neither has been published to date. The algorithms oftransreal arithmetic are explained, particularly clearly, in a recent treatment,in which transcomplex numbers are also introduced [5].

We develop a paraconsistent logic by expressing the Sheffer Stroke [9] as anoperation in transreal arithmetic. The Sheffer Stroke can be used to develop allclassical, truth functional logics [9] (See entry “Sheffer Stroke”), [28] (p. 29) soour paraconsistent logic is just one example from an entire class of paraconsistentgeneralisations of classical logic. During the development of our logic we explainthe motivation for each of the design decisions and consider alternatives so asto assist readers in developing their own paraconsistent logics that use transrealarithmetic as a consistent basis.

The Sheffer Stroke is better known, in Electronic Engineering, as the Not-And operation, NAND [28]. It can be used to develop all of the logic circuitsin digital computers. By generalising the NAND gate to paraconsistent form,we create the possibility of fabricating paraconsistent processors in hardware.We briefly consider features of high-level, computer languages that implementparaconsistent logics. This may be of interest to practitioners of Artificial In-telligence and Cybernetics (control theory).

2 Paraconsistent Logic

2.1 Truth Values

The transreal numbers are just the real numbers augmented with three non-finite numbers: negative infinity (−∞), positive infinity (∞) and nullity (Φ).Nullity is absorptive over the elementary arithmetical operations so that whenit is involved in a sum, difference, product or quotient, the result is nullity.However nullity is not universally absorptive, it may be an element of arbitrarymappings. Nullity is the only unordered number in transreal arithmetic [8][7].

Nullity’s absorptive properties make it a good candidate for a Gap valuethat has no degree of falsity or truthfulness.

The utility of a Gap value can be illustrated with the well known fairy storyof Goldilocks and the Three Bears. Goldilocks is interested in predicates con-cerning porridge. She wants to know the truth values of too cold(porridge),too hot(porridge), just right(porridge). But suppose Goldilocks’ reasoning sys-tem is presented with the logical argument porridge, devoid of a predicate. Whatis she to do? The argument porridge is a signifier for boiled oats that lie in abowl in front of her. Neither the signifier nor the actual boiled oats are in theclass of things that can be assigned degrees of falsity or truthfulness. Reflectingon her difficulty, Goldilocks may decide to assign a Gap value, that is a valuewith no degree of falsity or truthfulness, to the signifier porridge. Going furthershe may decide to apply Gap values to all badly formed formulas of logic andto everything that cannot be assigned a degree of falsity or truthfulness, suchas actual boiled oats. This makes Goldilocks’ reasoning system total. She canassign a logical value to every possible sentence, including meta sentences thatconsider the properties of actual objects, such as boiled oats, being presenteddirectly to her reasoning system. Notice that we must say “sentence” here, not“predicate,” if we are to admit badly formed formulae. We hope we have doneenough to convey the utility of a Gap value to the reader.

We define that negative infinity is classical False and positive infinity isclassical True. This has the merit that we have now used up all of the non-finite, transreal numbers, leaving all of the real numbers to convey dialetheaicdegrees of falsity and truthfulness.

In the next subsection we use arithmetical negation (unary subtraction) tomodel logical negation, thereby exploiting the following idempotence of transrealarithmetic: −(∞) = −∞,−(−∞) = ∞. Many, perhaps most, paraconsistentlogics will use some kind of idempotent negation but, perhaps, one that cyclesthrough many truth values, not just two. But not all logics have an idempotentnegation. For example many computer languages use zero for False and any non-zero value for True. This is exploited in cases, such as memory management,where an applications programmer instructs the assignment of memory and isreturned a single value by the operating system. If the returned value is non-zero, it is True that the operating system has assigned memory and the returnedvalue is the base address of that memory but if the returned value is zero thenit is False that the operating system has assigned memory. In such languages

the negation of any True value is the unique False value but the negation ofthe unique False value is exactly one, fixed one, of the True values. In thiscase negation is not idempotent on the truth values but it is idempotent on theclasses from which the specific truth values are drawn. One might even wantcontinuous forms of negation implemented, say, as rotations in the complexplane, or one might want continuous blends of negation with other operators[4]. Yet other kinds of negation might be wanted.

If we were to model logical negation with the arithmetical reciprocal thenwe might chose to model False with zero and True with infinity so that negationis supplied by the transreal idempotence 1/0 = ∞, 1/∞ = 0. This approachis taken by Gomide [18]; while logically sufficient, the operators are not totalfunctions of the transreal numbers, which complicates the application of ad-vanced mathematics to such a system. The computation of reciprocals also hasmore numerical error or higher computational cost than the arithmetisation usedhere. Of course logical negation can be modelled in many other arithmeticalways, including by arbitrary mappings.

If a fuzzy or statistical system is wanted then we work with probabilities, inwhich case transreal numbers outside the range from zero to one, inclusive, ariseonly in the underlying frequencies. Recall [17][15] that a statistical frequency,f = o/e, is the ratio of the number of target outcomes, o, to the number ofexperiments, e. This admits all non-negative, rational frequencies. The non-negative, irrational frequencies are admitted via probability density functions.The non-negative, non-finite, transreal numbers are also valid frequencies. Nul-lity, Φ = 0/0, is the frequency where no target outcomes have occurred in noexperiments, say where a coin is held on its edge, between two leaves of a table,before it is tossed, in an experiment to determine whether heads lie face-up onthe coin when it comes to rest. Infinity,∞ = k/0 = 1/0, for all positive k, is thefrequency where a positive number of outcomes, say one, occurs in no experi-ments, say where the frequency of heads is wanted in a coin tossing experiment,where the coin lies heads-up on a table before it is tossed for the first time.

These examples might strike the reader as artificial but they are needed tomake statistics total. The reader might be more satisfied by an example thatcommonly occurs in statistical packages: what is the arithmetical mean of a listof no numbers? Suppose the number of elements, in the list, is accumulated inthe variable n and the sum is accumulated in the variable s then the mean iscomputed as s/n. We program defensively by setting the accumulators to aninitial value of zero: n = s = 0. When the program handles an empty list, nand s are not incremented, which correctly records that there are no, that iszero, elements in the list and that they have no, that is zero, sum. The programthen computes the mean as n/s = 0/0 = Φ. No special handling of the emptylist is required, no matter what complexity of statistical computation follows.Thus transreal arithmetic simplifies computer code, removes all arithmeticalexceptions from syntactically correct, transarithmetical sentences and providesa consistent basis for fuzzy and statistical, paraconsistent logics.

Returning now to our paraconsistent logic, we define that the real numbersencode degrees of both falsity and truthfulness. The negative real numbers are

more False than True, the positive, real numbers are more True than False,zero is equally False and True. We relate the degree of falsity and truthfulnessmonotonically to the number modelling the truth value so that negative infinityis entirely False, that is classically False, and positive infinity is entirely True,that is classically True. The reader might want to impose specific monotonicfunctions so as to obtain particular metrics. In this regard the transreal arct-angent function [7] might be helpful because it maps a linear, transreal angleonto nullity and the whole range of transreal numbers from negative infinityto positive infinity. Nullity is the unique Gap value. Thus all transreal num-bers are used in our paraconsistent logic, making it both total and consistent.(The reader, being educated in logical terminology, will not confuse total andconsistent with complete and consistent.)

2.2 Metalogical Principle of Monotonicity

We have a metalogical intuition that a conclusion can depart no more from beingequally False and True than the most divergent of its premises. No matter howan inference is constructed, this is sufficient to cut off all those conclusions thatare more divergent so the inference is generally non-explosive.

In our model the absolute value of the conclusion can be no greater thanthe greatest absolute value of its premises. We operate on absolute values,not signed values, to allow the possibility of a conclusion being the negationof some one or more of its premises. Indeed we are forced into this strategyby implementing our paraconsistent logic in terms of the Sheffer Stroke whichinvolves a negation.

Enforcing metalogical monotonicity on all logical operators is a very simpleway to make a logic paraconsistent.

2.3 Sheffer Stroke

It is known that the truth functional (Boolean) operators for logical negation(not, ¬), logical conjunction (and, &), and logical disjunction (or, ∨) are func-tionally complete [9] (See entry “Sheffer Stroke” ), [28] (p. 29) so that anytruth functional operators can be derived from these three. In fact it is knownthat the sets {¬,&} and {¬,∨} are each functionally complete but it serves ourpurpose better to consider the wider set of operators {¬,&,∨}.

We begin by introducing the transreal minimum and maximum functions,which we use to define paraconsistent versions of the classical negation, con-junction and disjunction operators. We use negative infinity (−∞) to modelclassical False (F) and positive infinity (∞) to model classical True (T). We usenullity (Φ) to model the logical Gap value (G). Note that only the real num-bers model dialetheaic truth values. The three non-finite numbers each model asingle truth value: −∞ models classical False,∞ models classical True, Φ mod-els Gap. We then prove that the paraconsistent operators contain the classicalones. With a little extra work we prove that the paraconsistent operators arewell defined for all transreal arguments when we assume that the finite, truth

values are arranged monotonically with the real numbers that model them. Wethen define a paraconsistent version of the Sheffer Stroke (|). There are three,well known, identities that relate the classical Sheffer Stroke to classical nega-tion, conjunction and disjunction. We show that these identities hold when wesubstitute the paraconsistent Sheffer Stroke and the paraconsistent negation,conjunction and disjunction. Thus we prove that the paraconsistent operatorsare defined everywhere and are consistent with their classical counterparts.

We begin by defining the binary, transreal, minimum and maximum func-tions so that the minimum of two transreal numbers is the least, ordered oneof them or else is nullity. Similarly the maximum of two transreal numbers isthe greatest, ordered one of them or else is nullity. These definitions rely onthe three transreal relations less-than, equal-to, greater-than as axiomatised in[8] and explicated in [6]. It is sufficient for the reader to know that: nullityis the uniquely unordered, transreal number so it is the only transreal numberthat compares not-less-than, not-equal-to and not-greater than any other dis-tinct number; negative infinity is the least, ordered, transreal number; positiveinfinity is the greatest, ordered, transreal number.

Definition 1. Transreal minimum,

min(a, b) =

a : a < ba : a = ba : b = Φb : b < ab : a = Φ

.

Definition 2. Transreal maximum,

max(a, b) =

a : a > ba : a = ba : b = Φb : b > ab : a = Φ

.

The minimum and maximum functions, just defined, treat nullity non-absorptivelybut we chose to treat the logical Gap value absorptively.

Definition 3. Paraconsistent conjunction,

a & b =

{Φ : a = Φ or b = Φmin(a, b) : otherwise

.

Definition 4. Paraconsistent disjunction,

a ∨ b =

{Φ : a = Φ or b = Φmax(a, b) : otherwise

.

We now define the paraconsistent, logical negation as transarithmetical nega-tion.

Definition 5. Paraconsistent negation, ¬a = −a.

Transreal arithmetic has −0 = 0, −Φ = Φ and in all other cases, the negationis distinct so that −a 6= a.

The Sheffer Stroke (|) may be defined as an infix operator but we follow themore modern practice of taking it as a post-fix operator so that no bracketingis needed. This leads to shorter and clearer formulae.

Definition 6. Paraconsistent Sheffer Stroke, ab| = ¬(a & b), with all symbolsread paraconsistently.

We now prove that the paraconsistent negation, conjunction and disjunctioncontain their classical counterparts and that the paraconsistent operators arewell defined for all transreal arguments.

Theorem 7. Paraconsistent negation contains classical negation.

Proof. Classical negation has ¬F = T and ¬T = F. Equivalently paraconsistentnegation has ¬(−∞) = −(−∞) =∞ and ¬∞ = −∞.

Theorem 8. Paraconsistent conjunction contains classical conjunction.

Proof. Classical conjunction has F & F = F; F & T = F; T & F = F; T & T = T.Equivalently paraconsistent conjunction has −∞ & −∞ = min(−∞,−∞) =−∞; −∞ & ∞ = min(−∞,∞) = −∞; ∞ & − ∞ = min(∞,−∞) = −∞;∞ & ∞ = min(∞,∞) =∞.

Theorem 9. Paraconsistent disjunction contains classical disjunction.

Proof. Classical disjunction has F ∨ F = F; F ∨ T = T; T ∨ F = T; T ∨ T = T.Equivalently paraconsistent disjunction has −∞ ∨ −∞ = max(−∞,−∞) =−∞; −∞ ∨ ∞ = max(−∞,∞) = ∞; ∞ ∨ −∞ = max(∞,−∞) = ∞;∞ ∨ ∞ = max(∞,∞) =∞.

Theorem 10. Paraconsistent negation, conjunction, and disjunction are welldefined for all transreal arguments.

Proof. Paraconsistent negation, conjunction, and disjunction are defined for alltransreal arguments. It remains only to show that these operators are mono-tonic. Firstly nullity is absorptive in these operators so that if any argument isnullity the result is nullity. Nullity is disjoint from all other transreal numbersbecause it is the uniquely isolated point of transreal space [7], therefore nullityresults are disjoint from all other transreal results and cannot contradict them.Secondly the preceding three theorems show that the paraconsistent operatorsare well defined at the boundaries −∞ and ∞ but, by definition, the non-nullity, paraconsistent, truth values are monotonic so the operators just definedare monotonic for all transreal t in the range −∞ ≤ t ≤ ∞. This completes theproof for all transreal arguments.

We now derive the paraconsistent negation, conjunction and disjunction fromformulae involving the paraconsistent Sheffer Stroke. This proves that the para-consistent Sheffer Stroke is functionally complete both for classical truth valuesand for the paraconsistent truth values defined here.

Theorem 11. pp| = ¬p for all transreal p.

Proof. pp| = ¬(p & p) = ¬p, with all symbols read paraconsistently.

Theorem 12. pq|pq|| = p & q for all transreal p, q.

Proof. pq|pq|| = (¬(p & q))(¬(p & q))| = ¬(¬(p & q)) = p & q, with all symbolsread paraconsistently.

Theorem 13. pp|qq|| = p ∨ q for all transreal p, q.

Proof. pp|qq|| = (¬p) (¬q) | = ¬((¬p) & (¬q)) = p ∨ q by the classical deMorgan’s Law, generalised to all transreal numbers by monotonicity and theabsorptiveness of nullity, with all symbols read paraconsistently.

2.4 Logical Space

Figure 1: Two Bands of the Possible Worlds Rainbow

Wittgenstein discusses the concept of a logical space [16]. We now constructa transreal space of all paraconsistent, possible worlds whose predicates take ontransreal truth values, including subsets such as real and Boolean truth values.

Firstly we note that the set of transreal numbers {t : −∞ ≤ t ≤ ∞, t = Φ}can be mapped onto, t′, a line segment, with a length of one half of a unit,augmented by the point at nullity, using the transreal arctangent [7] as t′ =arctan(t)/2π. Thus all of the transreal, truth values for a single predicate canbe mapped onto a half-unit line-segment, augmented with the point at nullity.

Secondly we list the countable infinitude of predicates, pi, in some fixedorder, with i running over the strictly positive integers. We then lay off thetruth values of the ith predicate in a half-unit segment from i to i + 1/2, withthe point at nullity at i+ 3/4. A possible world has exactly one transreal, truthvalue or, equivalently, exactly one geometrical point, in the ith union of the ithline segment and ith point at nullity. Thus points on a half-infinite line encodesone possible world.

Thirdly we rotate the half-infinite line by a full rotation and index eachradius by a real angle, θ, in the range 0 ≤ θ < 2π. As each distinct radius isa distinct possible world, we have constructed a continuum of possible worldswhose cardinality is just sufficient to model the set of all possible worlds. Tosee the sufficiency of the cardinality note that we may select the truth valuesin one line segment with the cardinality of the continuum but the selections inthe other line segments do not increase the cardinality [32].

The figure just constructed, see Figure 1, is a concentric, alternating se-quence of an ith annulus and an ith circle, with an empty unit disc at thecentre. We call the union of an annulus and its circle a “band” and we refer tothe whole figure as a “rainbow” in anticipation of variants that are taken, notover a full rotation, but over an arc.

Many variations on this construction are possible. We might want to: letthe angle range over more than a full rotation so that the windings encode acountable infinitude of partitions of all possible worlds; let the angle range oversome quantity other than an integral number of full rotations so that both thelower and upper bounds of the range are included in the range; take the angleabout zero so that it can be constructed by a Lie group [1]; take the angle inthe range −π/2 ≤ θ ≤ π/2 or θ = Φ so that it is in the principal range ofthe transreal arctangent [7]. We might want to define some other planar shapeentirely.

2.5 Partitioning, Significance and Consistency

One way to handle inconsistent predicates is to partition them into internallyconsistent partitions [26]. Da Costa [13] uses a hierarchy of such partitions. Thepartitions may be constructed syntactically or by semantic relevance [19][25][27].Self evidently this produces islands of consistent reasoning but it does more.Priest [26] discusses the well known preface paradox in which an author writesa book containing many statements, which the author has good grounds tobelieve are true, but who also writes, in the preface, that in any list of manystatements there must be errors. Thus the author is in the paradoxical, orparaconsistent, position of believing every single one of the statements andnot believing, actually disbelieving, the conjunction of all of the statements.

This paradox is dissolved by Williams [33] who points out that a human canrationally believe a set of statements and not believe their conjunction. He citesthe example of a motorcycle rider who knows a certain route that involves aleft-hand bend followed by a right-hand bend. The rider successfully negotiatesthe route. Over time the rider comes to learn the conjunction that the left-handbend is followed by the right-hand bend and changes behaviour to take the faster‘racing line’ through the switchback. Thus it is human experience that one canknow many propositions and not know their conjunction. Hence there is nomystery in the fact that a human can believe inconsistent predicates if he or shehas not derived a contradiction. This issue is taken up in a description of howautoepistemic reasoning can be implemented in a finite (Prolog) implementationof a system’s own beliefs [22]. Thus we have reason to expect that human andall finite reasoning may contain inconsistent statements and that one way tohandle them is via partitioning.

There is a simpler way to handle inconsistencies using arithmetic. Supposewe have the sentence “p & ¬p.” If we handle this sentence algebraically we mayconclude, by conjunction elimination, that p is True and ¬p is True, regardlessof whether the truth values are classical or paraconsistent. But if we evaluatethis proposition arithmetically we have either T & F = F or F & T = F,the conjunction fails, so we cannot derive a contradiction. Hence reasoningarithmetically blocks the derivation of contradictions but reasoning algebraicallymay admit them. It is no surprise, then, if human reasoning has evolved to beparticular, working on concrete mental models, rather than having great facilityin general reasoning [21].

Suppose we reason arithmetically and keep a record of the serial or parallelsequence of points we visit in a logical space. Then the trace is necessarilyconsistent because it was derived arithmetically. In a biological system, wemight reinforce all of the neurones in the trace [14]; in a computer system westore the trace as a collection of line segments. In paraconsistent terms thetrace is a partition. If we find more than one line segment passing through apoint then all of the traces are consistent at that point and we may combinethe common parts of the traces. The common parts form a partition that havebeen consistent over the entire history of the arithmetical reasoning system.

If all of the components on an axis in logical space are negative then all ofthe system’s history supports the conclusion that the predicate tied to the axisis more False than True; if positive then the predicate is historically more Truethan False; if there are components that are both positive and negative thenthe predicate is historically contradictory or paraconsistent. Hence a reason-ing system that has sufficient geometrical or algebraic power may detect theinconsistency or paraconsistency. If many traces haven components in this axisthen a contradiction or paraconsistency is important to the mental life of thesystem. This bears on a case discussed by Priest [26]. Bohr’s model of the atomis useful in predicting atomic structure and is, in our terms, associated withmany traces. Maxwell’s equations are useful in electrodynamics and are alsoassociated with many traces. But the two are inconsistent in that the orbit ofa Bohr electron should decay due to Maxwell radiation, which it does not. The

non-decay is therefore significant, because it is associated with many traces,and finds resolution in the combination of quantal energy levels and the Pauliexclusion principle. Thus logical space offers a measure of the significance ofa predicate, in terms of the number of traces that pass through it in logicalspace, and we now arrive at a measure which classifies classical and dialetheaicconsistencies. Let our desideratum be d = p & ¬p. If d = −∞ then predicate pis a classical consistency; if −∞ < d < 0 then p is a dialetheaic consistency; ifd = 0 then p is both a dialetheaic consistency and a dialetheaic contradiction;if 0 < d < ∞ then p is dialetheaic contradiction; if d = ∞ then p is a classicalcontradiction; if d = Φ then p is a Gap.

2.6 Advanced Paraconsistent Logics

We have deliberately introduced an elementary paraconsistent logic. It is juststrong enough to generalise propositional logic [23] to paraconsistent form butthis means it is a basis for generalising more advanced paraconsistent logics. Forexample the computer language, Prolog [12], provides a propositional logic [23]with existential and universal quantifies introduced by controlling the scope ofvariables. Variables with scope local to a clause are existential, while variableswith global scope are universal [31] (p. 10). Any propositional logic is triviallyextended by the paraconsistent logic presented here but the example of Prolograises a deeper issue. Prolog works by unifying variables with arguments. Theunification triggers execution of the clause but in a paraconsistent logic, such asours, all paraconsistent truth values will unify with arguments to some extent.One way to handle this would be to allow all clauses to execute in parallel butit is unlikely that a practical machine would have the resources to do this formany clauses. A more practical approach might be to allow only the execution ofclauses that have a high degree of truthfulness, embedding a best-first strategyin the execution of the computer language. Triggering on truthfulness reflectsthe human bias for positive reasoning but in an artificial intelligence we mayprefer to trigger on a high degree of divergence from being equally False andTrue. In a practical system we might want some measure of usefulness, expectedvalue [17], or subjective value [14] to drive the selection of clauses, in which casewe could couple a value parameter with clauses. This takes us deeply into thedesign of robots and the modelling of animal and artificial intelligences.

The matching of all paraconsistent truth values to a clause’s variables leadsus to expect that useful paraconsistent logics will have some implicit or explicitnotion of execution control, raising the prospect that they should be as power-ful as the Turing machine. Clearly paraconsistent logics can be very powerful,taking them a very long way from the elementary, paraconsistent logic intro-duced here. Our paraconsistent logic may be of interest precisely because ofits mathematical simplicity. We invite the reader to consider if more compe-tent, paraconsistent logics, would benefit from being arithmetised in transrealarithmetic rather than calling on the advanced, even heroic, mathematics thatis usually used as a consistent basis for a paraconsistent logic.

3 Conclusion

We present a paraconsistent logic by arithmetising the Sheffer Stroke in tran-sreal arithmetic. We use negative infinity to model classical False and positiveinfinity to model classical True. The real numbers model truth values that areboth False and True. The negative, real numbers are more False than True; thepositive, real numbers are more True than False; zero models the truth valuethat is equally False and True. The transreal number nullity, which is unorderedand lies outside the range of ordered numbers from negative infinity to positiveinfinity, models truth values that have no component of falsity or truthfulness.These Gap truth values are neither False nor True. As the Sheffer Stroke canbe used to develop all classical, truth functional logics, our logic is just one ofa class of paraconsistent generalisations of classical logic. The reader, who isskilled in formal logic, can easily develop other paraconsistent logics using tran-sreal arithmetic as a consistent basis. This is a very much simpler approach tothe formalisation of paraconsistent logics than is usually attempted so it bringsparaconsistent logics within the reach of non-specialists, such as Electronic En-gineers and Computer Scientists, who have been trained in classical, Booleanlogic.

The Sheffer Stroke directly implements the operation Not-And (NAND).NAND gates can be used to implement all of the logic in general purpose com-puters so our paraconsistent generalisation of the Sheffer Stroke specifies thebehaviour of a paraconsistent NAND gate. This could be used to implementparaconsistent processors in hardware. Such hardware might be particularlywell suited to the execution of Artificial Intelligence and Cybernetics programs,for example those which exploit models of neural processing or fuzzy logic.

We give a geometrical construction which encodes all paraconsistent, possibleworlds in an alternating sequence of an annulus and a circle that fills out thereal plane, with each radius marking off one possible world.

We point out the obvious property that evaluation of logical sentences in-volving only connectives and literal False and True values, not variables, cannotsupport the conclusion of a contradiction. Thus on-the-fly evaluation of a lit-eral sentence is always consistent. We propose that this property can be usedto partition predicates into internally consistent sets and that this provides ameasure of the significance of predicates. We also give a simple, transarith-metical measure, a desideratum, d = p & ¬p, for categorising predicates intoclassical or else dialetheaic form and for categorising classical or else dialetheaicconsistencies and contradictions.

We give a very simple principle of metalogical monotonicity that forces alogic to be non-explosive. In future we might examine existing paraconsistentlogics to see if they implicitly obey this principle.

Given the interest there is in the number nullity, perhaps the single mostimportant, original contribution of this paper is to observe that the transrealnumber nullity provides a faithful model of Gap values in advanced logics.

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